Dropping down to dlvl 75, I ended up in a small rectangular room. Casting my usual mae, mdf and mcg, I saw no traps, no monsters and only one unknown item in the room. I sauntered over to take a look and saw a Mithril ring. At this point there were only 5 unknown rings, 4 of power and Tulkas. A Mithril ring sure sounded like...yes! Nenya! I can't believe I found it without having to fight my way through a horde of Pit Fiends. Now I can get rid of this shabby looking Hard Leather Cap of Thranduil and finally wear this beautiful Golden Crown of Gondor that I've been hauling around to occasionally protect me from confusion and stunning. I can also lose the Ring of Constitution <+6> that I have been wearing forever, it seems. And my Ring of Speed <+12> is pretty much redundant at this point. Not much difference between speed 34 hasted and speed 46 as far as I can see. This opens up a ring slot and hence my question.

I have two candidate rings, a Ring of Accuracy (+21,0) and a Ring of Damage (0,+15). I see that accuracy is dismissed a lot here, but looking in my monster memory at the Great Wyrm of Chaos, for instance, I see that by wielding the Ring of Accuracy, my chance to hit rises from 35% to 47%. When wielding the Ring of Damage, my sword does 348 damage per round against evil creatures. Multiplying this by 0.35 yields effective damage per round of 121.8. Meanwhile, wielding the Ring of Accuracy gives 292 damage per round. Multiplying by 0.47 yields 137.2 effective damage. The Ring of Accuracy has the additional benefit of fewer wasted shots when I use my bow. So how is it not advantageous to choose the Ring of Accuracy over the Ring of Damage?

Your chance to hit sounds abysmal. Most characters have a chance to hit in the upper 60th percentile at minimum, and a Ring of Accuracy gives only 1-4% improvement in hit chance, not the 12% you're seeing.

So perhaps it is conditional then. I'm normally a mhh sort of guy, but when it comes to dragons, I like to get up close and personal. I have observed that when they can use their teeth and claws, they are a lot less likely to breathe all sorts of nasty stuff at you.

The advice to choose Damage over Accuracy is, as jevansau noted, intended for characters capable of melee combat, and who have a lot of blows. As a mage you get 2/3 the advantage a warrior gets from a ring of damage, and your base accuracy is low enough that improving it has actual benefits. That said, watch out. Your character should never fight a Great Wyrm of Chaos or anything that has a similar AC in melee. These calculations would be better made on the type of monster you do end up fighting in melee as an end-game mage, who tend to have much lower AC, meaning your base accuracy is higher, meaning that increasing accuracy is probably less necessary/helpful.
The advice is also kind of a holdover from back when the effect of increased damage was immediately apparent, while the effect of increased accuracy was invisible.

Traditionally I would recommend using a different ring on a mage, except you have absolutely everything including stats, resists, abilities, speed covered quite nicely so you may as well go ahead and use the Accuracy ring. Just don't let it tempt you into actually using melee against anything even remotely threatening, whether it needs to die fast or is scary in melee.

EDIT: Written before I saw your post. Unless something has recently changed, dragons are in fact equally likely to breathe at you at all times. They are, on the other hand, going to do a lot more damage with their breaths in melee range than they would, say, 10 tiles away, since breath attenuation was implemented some time back. Note that other monsters, including Sauron and Morgoth, will also use spells at the same frequency in and out of melee. The only significant difference being in melee causes (outside of effects on breath damage) is that when their AI rolls not to cast a spell, they will hit you if in melee range and move towards you if not.

Hmm, this is not what I have observed. When fighting dragons from a distance, they almost always breathe. When fighting at melee range they rarely breathe. It takes a long time to whittle them down with so little damage per round, but @ can take melee damage with 252 AC after mee, since they tend to miss often. So far @ has defeated 15 Great Wyrms with this strategy. Of course, I am using melee mainly to preserve mana here, so I do tend to throw in a few Mana Storms to help with the whittling.

Another advantage of melee is that it tends to keep neighboring creatures asleep. As a Hobbit @ has enough stealth to be safe from anything that is not permanently awake, such as Zephyr Hounds and Quylthulgs. In fact, @ needs to walk up and bop a Great Wrym on the nose to get its' attention.

It is possible having a lot of AC helps make the melee damage of a Great Wyrm less scary (they are not particularly powerful), but it still stands that they are just as likely to breathe (I am quite certain), and their breaths do full damage, rather than the damage being reduced (of which I am entirely certain). Additionally, by taking dragons down slower, you give them a lot more time (which they can use to breathe) and because you are doing damage slowly, they remain capable of doing a lot of damage with their breaths for longer.

There are a lot of cognitive biases that come into play when playing angband, which has a lot of unintuitive probabilities and emotional baggage (you notice the third consecutive failed teleport a lot more when it makes you die etc.).

I don't know much about how monster awakeness works, and it's possible something changed with the tweaks to pathfinding and such, but last I recall, the only activities that really affected how monsters wake up or not was whether you had damaged them or shone a light in their direction. I doubt that melee is significantly quieter (especially when you factor in the extra time killing the monster takes) than spellcasting, in any case.

As for conserving mana, yeah, that's always nice to do. I recommend relying a bit more on devices to supplement mana, mages get a lot of bonus damage with those in current V. A wand of annihilation with a mage can consistently do around 300 damage a round, at range, with the only mana cost being recharging. Even rods would be likely to outdamage your melee, I think.

I gave up on wands a long time ago. I had a Wand of Drain Life that I tried on a monster that had 840 HP and it only took it down 1 star per hit which I assume is 84 HP. This in spite of the fact that it claimed to do 150 HP damage and my device skill would raise that by some large percentage. I do have a Wand of Annihilation in my home that claims to do 250 HP damage and my device skill will currently raise that by 71 percent for presumably 427.5 HP damage. I'll give it a try to see if that is correct or not.